Irradiance

What irradiance means

Power tells you how much energy a source delivers each second. But a beam of light spread over a big area feels weaker than the same beam squeezed onto a small one. To compare fairly, we share the power over the area it falls on.

Notice that the same power on half the area doubles the irradiance. This is why a magnifying glass can set paper alight: it does not create energy, it concentrates it.

Once you have the irradiance, you can also work backwards. Power is $P = IA$, and energy is $E = Pt$. In everyday life energy is often given in kilowatt-hours: $1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$.

Efficiency and solar farms

A solar panel does not turn all the incident power into electricity. Efficiency compares the useful output with the total input:

The power in is $P = IA$. So the electrical power is $P_{out} = \text{efficiency} \times IA$.

Why is the average irradiance so much lower than $800\ \text{W m}^{-2}$ on a clear day? Because the average includes nights, cloud and low winter sun.

Distance: the inverse-square law

Stand beside a lamp and it is bright. Walk away and it dims quickly. The lamp's total power has not changed, but it now spreads over a bigger area.

A point source emits equally in all directions. At distance $r$ the energy is spread over a sphere of area $4\pi r^2$, so

Doubling the distance makes the area 4 times bigger, so the irradiance falls to a quarter. This is an inverse square law: $I \propto \dfrac{1}{r^2}$.

This is also how you verify the law with a light meter: measure $I$ at several distances $d$, and show that $Id^2$ is the same each time, or that a graph of $I$ against $1/d^2$ is a straight line through the origin.

Sunlight, angle and focusing

The Sun is a point source far away. Its power is $3.8\times10^{26}\ \text{W}$ and the Earth is $1.5\times10^{11}\ \text{m}$ away, so at the top of the atmosphere

This is the solar constant, about $1{,}400\ \text{W m}^{-2}$.

The Earth intercepts sunlight over a circle of area $\pi R^2$ (a disc), but its surface is a sphere of area $4\pi R^2$. So the average over the whole surface is a quarter of the solar constant, because the same intercepted power is spread over a surface four times larger (this accounts for night and slanted areas).

Tilted surfaces

If rays strike a surface at an angle $\theta$ to the normal, the same beam is spread over a larger area, and the effective irradiance is

This is why Irish panels face south and are tilted, and why winter sunlight (low in the sky) gives less than summer sunlight. Cloud and atmosphere also absorb and scatter, so the irradiance on the ground in Ireland is below the solar constant.

Focusing

A lens gathers the power from a wide beam onto a tiny spot. The power is the same but $A$ shrinks, so $I$ grows enormously. Since area scales with the square of the diameter, reducing a beam's diameter by a factor of 100 raises irradiance by $100^2 = 10^4$. The lens of the eye does exactly this to light entering the pupil, focusing it onto a tiny patch of the retina. That is why even a low-power laser can damage an eye.

Sources across the electromagnetic spectrum

Irradiance is measured for any electromagnetic (EM) radiation, not only light. All EM waves travel at $c = 3.0\times10^{8}\ \text{m s}^{-1}$ in a vacuum and obey $c = f\lambda$. Each region has typical sources and uses.

Moving from radio to gamma, frequency and photon energy increase and wavelength decreases. The highest-energy photons are in the gamma region. Ultraviolet, X-rays and gamma rays are ionising: their photons carry enough energy to remove electrons from atoms and can damage living cells.

Vuelve al principio: a panel in Cork receives a lower irradiance than one in Athens because the Sun is lower in the sky (a larger angle $\theta$ and a longer path through the atmosphere), and there is more cloud. Same panel area, less $P = IA$.

Summary

• Irradiance is power per unit area: $I = P/A$, in $\text{W m}^{-2}$. Then $P = IA$ and $E = Pt$ (use kWh for household energy).

• Useful output is efficiency × input power, so a solar farm produces $\text{efficiency} \times IA$.

• For a point source, $I = P/4\pi r^2$: irradiance follows an inverse square law, $I \propto 1/r^2$.

• On a tilted surface the effective irradiance is $I\cos\theta$, with $\theta$ measured from the normal.

• Focusing reduces area and so raises irradiance greatly, which is why laser beams can harm the eye.

• Across the EM spectrum from radio to gamma, frequency and photon energy rise and wavelength falls; UV, X-rays and gamma rays are ionising.